Random walk conditioned to stay positive

Abstract

Given a random walk with mean zero and finite variance,
the progress to infinity of
the associated random walk conditioned to stay positive is
studied through the sample path representation of Tanaka
(1989). Specifically, if $D(x)$ is the time that
the process spends below $x$ and $\phi(x)= \log \log x$ then,
as $x$ goes to infinity,
$D(x) / x^2 $ ultimately lies
between $L / \phi(x)$ and $U \phi (x)$ for
suitable (non-random) positive $L$
and finite $U$. The Bessel-3 is one continuous
analogue; for it the best $L$ and $U$ are identified.